Equivalents of the Partition Principle

Question

The Partition Principle (PP) is a weakening of AC stating that if A surjects onto B then B injects into A.

It is well-known that PP implies the dual Schroeder-Bernstein theorem (SB*) by invoking SB. As well, it is sufficiently strong to prove AC$_{\kappa}$ for all $\aleph$ numbers $\kappa$. Because of this it proves DC and hence the existence of a Lebesgue non-measurable set. See also the proof by Sierpinski in:

Sierpinski, W. "L’axiome de M. Zermelo et son rôle dans la théorie des ensembles et l’analyse." Bulletin de l’Académie des Sciences de Cracovie, Classe des Sciences Math., Sér. A (1918) (1918): 97-152.

However there are few of any well-known (to my knowledge) equivalents of PP. What are some known equivalent results (over ZF), if any? If there aren't, what are the particular challenges associated with finding statements equivalent to PP?

Answer 1

One easy equivalent is "multiplication is repeated summation" for cardinals. In other words:

For any $\{A_i\mid i\in I\}$ there is an injection from $\bigcup A_i$ into $\bigcup\{i\}\times A_i$, i.e. a union is always less or equal to its disjoint union.

This follows from $\sf PP$ since the disjoint union easily maps onto the union by the projection map; and it implies $\sf PP$ since if $f\colon A\to B$ is a surjection, then let $B_a=\{f(a)\}$, then $B=\bigcup_{a\in A}B_a$ and $|A|=|\bigcup_{a\in A}\{a\}\times B_a|$, the latter is the disjoint union of course.

Another, which is perhaps just a reformulation of $\sf PP$,

If $R$ is a relation, then $|\operatorname{dom}(R)|\leq|R|$.

This is rather trivial, since if $f$ is a surjection, let $R$ be $f^{-1}$ (the inverse relation, not an inverse mapping!) and we immediately get $\sf PP$.

Now, are there more? Probably. You can probably come up with variants of these arguments. But it's not going to shed a lot of information, at the end. It seems to me that $\sf PP$ is so close to being equivalent to $\sf AC$, that it's hard to separate them without better tools for constructing models of $\sf ZF$, or rather $\sf ZF+DC$, since we know that $\sf DC$ must hold if we are to satisfy $\sf PP$. I have some ideas, but it's still to early to talk about whether or not they will work. Maybe in 2030.